
theorem Th22:
  for X be RealNormSpace for s be sequence of X holds s is
  summable implies for n be Nat holds s^\n is summable
proof
  let X be RealNormSpace;
  let s be sequence of X;
  defpred X[Nat] means s^\$1 is summable;
A1: for n be Nat st X[n] holds X[n+1]
  proof
    let n be Nat;
    reconsider s1 = NAT --> (s^\n).0 as sequence of X;
    for k be Nat holds s1.k = (s^\n).0
     by ORDINAL1:def 12,FUNCOP_1:7;
    then
A2: Partial_Sums(s^\n^\1) = (Partial_Sums(s^\n)^\1) - s1 by Th21;
    assume s^\n is summable;
    then Partial_Sums(s^\n) is convergent;
    then
A3: Partial_Sums(s^\n)^\1 is convergent by Th7;
    s1 is convergent by Th12;
    then s^\(n+1)=(s^\n)^\1 & Partial_Sums(s^\n^\1) is convergent by A3,A2,
NAT_1:48,NORMSP_1:20;
    hence thesis by Def1;
  end;
  assume s is summable;
  then
A4: X[0] by NAT_1:47;
  thus for n be Nat holds X[n] from NAT_1:sch 2(A4,A1);
end;
